J. zyxwvutsr Fluid zyxwvutsrqpo Mech. zyxwvutsrq (1994), vol. 259, pp. 219-240 zyxwvut Copyright zyxwvutsrq 0 1994 Cambridge University Press 219 The role of coherent structures in bubble transport by turbulent shear flows By K. J. SENE’, J. C. R. HUNT’ AND N. H. THOMAS3 Institute of Hydrology, Wallingford, Oxfordshire OX10 8BB, UK Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK and Meteorological Office, Bracknell, Berkshire RG12 2SZ, UK FAST Team, Chemical Engrg and FRED Ltd, Research Park, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK (Received 31 January 1989 and in revised form 2 August 1993) Using Auton’s force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter 17 zyxwvutsr = AU2/2gx and a trapping parameter r = AUI VT where AUis the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies. A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference A U. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that r needs to have a value of about 3 to trap about 50 % of the bubbles if I7 is about 0.5 and greater if 17 is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as r and 17 increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars. 1. Introduction Observations of the motions of bubbles in air-entraining flows by Thomas (see Goldring, Mawer & Thomas 1980; also Thomas 1982) and in the wakes of bluff-body flows ( H u h , Fierfort & Coudol 1982) have revealed the phenomenon of bubbles travelling in discrete clusters within free shear layers. This behaviour (figure 1, taken from Goldring et al.) has been attributed to the presence of large-scale coherent eddies. In the present paper we seek to determine the influence of such coherent structures on the distributions of bubbles in free shear layers, using theoretical models and computer simulations. The plane coflowing turbulent mixing layer was chosen as a model field